On the Riesz Almost Convergent Sequences Space

Abstract

The purpose of this paper is to introduce new spaces 𝑓 and 𝑓0 that consist of all sequences whose Riesz transforms of order one are in the spaces 𝑓 and 𝑓0, respectively. We also show that 𝑓 and 𝑓0 are linearly isomorphic to the spaces 𝑓 and 𝑓0, respectively. The 𝛽- and 𝛾-duals of the spaces 𝑓 and 𝑓0 are computed. Furthermore, the classes (𝑓∶𝜇) and (𝜇∶𝑓) of infinite matrices are characterized for any given sequence space 𝜇 and determine the necessary and sufficient conditions on a matrix 𝐴 to satisfy 𝐵𝑅−core(𝐴𝑥)⊆𝐾−core(𝑥), 𝐵𝑅−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥) for all 𝑥∈ℓ∞.